Topological Realizations and Orbit Equivalente
The problem of realizing ergodic automorphisms of Lebesgue probability spaces as topological actions with certain properties has long been of interest in dynamics. In these two talks we will look at the speaker´s approach to this problem which involves realizations within a given strong orbit equivalence class as defined by Giordano, Putnam and Skau.
Iterated integrals and limit cycles
In this talk we review the theory of iterated integrals which appear in the study of limit cycles of planar vector fields.
In particular, we show that after a deformation of a holomorphic foliation with a generic first integral in the complex plane if the deformed holonomies of two certain vanishing cycles commute then the deformed foliation has also a first integral and so the deformed holonomies are identities.
Absolutely continuous invariant measures for non-uniformly expanding skew-products
We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicity along the leaves implies existence of absolutely continuous invariant probability measures. The main technical tool is an extension for sequences of maps of a result of de Melo and van Strien relating hyperbolicity to recurrence properties of orbits. As a consequence of our main result, we also obtain extensions of Keller’s theorem guaranteeing the existence of absolutely continuous invariant measures for non-uniformly hyperbolic one dimensional map.
Exponential mixing for the Teichmüller flow in the space of quadratic differentials
Masur and Veech proved that the Teichmüller geodesic flow is ergodic (with respect to the canonical invariant volume form) in each connected component of each stratum of the modul space of quadratic differentials. In view of the presence of an SL(2,R) action, ergodicity implies mixing in this context, and the question of the speed of mixing is thus naturally posed. Exponential mixing (for a class of weakly Holder observables introduced by Ratner) is equivalent to the spectral gap property for the SL(2,R) action.
Recently Avila, Gouezel and Yoccoz have shown exponential mixing for the Ratner class in the case of strata of squares, corresponding to the important case of Abelian differentials. We prove exponential mixing for the Ratner class in all strata and, in particular, for the action of the Teichmüller flow on the whole moduli space of quadratic differentials.
This is a joint work with Artur Avila.
Distance Expanding Random Mappings, Thermodynamic Formalism, Gibbs Measures and Fractal Geometry
Lyapunov exponents in nonuniformly hyperbolic dynamics
I will discuss some approaches to study orbits that lack uniform hyperbolic behavior. Such approaches can be used, for example to study level sets of Lyapunov regular points with equal exponent. If the dynamical system is not uniformly hyperbolic, then the set of points with zero Lyapunov exponent can be quite large or small (when measured e.g. in terms of fractal dimension or topological entropy).
This is investigated by means of the thermodynamic formalism for sub-systems which are uniformly hyperbolic. Such a scheme can be succesfully applied to primary examples of conformal dynamics such as parabolic or unimodal intervalo maps and rational maps on the Riemann sphere. However, principle techniques also extend to surface diffeomorphisms and certain flows in 3-dimensional manifolds.
Thermodynamical formalism of expanding measures
In this talk we will discuss some rich statistical properties of dynamical systems with a dense non-uniformly hyperbolic set: there exists an induced Markov structure that is adapted to all fat expanding measures, and there are finitely many "expanding" equilibrium states for every Hölder continuous potential. In the case of Viana maps we deduce that there exists a unique equilibrium state for every H"older continuous potential at high temperature. This is a joint work with Vilton Pinheiro.
Valores propios de sistemas minimales de Cantor de rango finito
En esta charla se mostrarán algunas ideas recientes en relación a la existencia de valores propios medibles y continuos en sistemas minimales de Cantor que se pueden representar con sistemas de Bratelli-Vershik donde aparecen un número acotado de torres.
Mechanisms for robust transitivity
I´ll explain the main ideas of an ongoing work with S. Crovisier in which we attempt to understand the geometric mechanism(s) which "explain" robust transitivity in the context of partially hyperbolic diffeomorphisms with 1-dimensional center bundle.
Invariant measures for interval maps with critical points and singularities
We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with in nite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.
Deformations of Benedicks-Carleson unimodal maps
We study smooth families of unimodal maps f_t which are inside a topological class (a "deformation"). Assuming that this topological class satisfies the so-called Topological Slow Recurrence, a (topological) condition that implies the more familiar Benedicks-Carleson and Collet-Eckmann conditions, we show that the families of conjugacies h_t satisfying h_t(f_0(x))=f_t(h_t(x)) is differentiable with respect to the parameter t. This is a work in progress with Viviane Baladi.
New criteria for residual hyperbolicity based in periodic sets.
We prove that if in the periodic sets of local C^1 diffeos in a residual subset of an open set a condition of non uniform hyperbolicity/non uniform expansion holds, then residually, hyperpolicity also holds residually for respective nonwandering sets.
Classification of $mathbb Z$ sofic shifts appearing as projective subdynamics in ${mathbb Z}^d$ shifts of finite type
Motivated by Hochman´s general notion of subdynamics for ${mathbb Z}^d$ shifts we define {it projective subdynamics} of ${mathbb Z}^d$ shifts of finite type (SFTs). Those are $k$-dimensional shifts, for some $k<d$, which can be seen inside a ${mathbb Z}^d$ SFT using projections onto lower dimensional lattices.
Restricting to one-dimensional projective subdynamics we present a complete classification of $mathbb Z$ sofic systems that can be realized inside ${mathbb Z}^d$ SFTs. We show that not all $mathbb Z$ sofics appear and that there is a distinction between {it stable} and {it unstable projective subdynamics}.
The presented results were obtained in collaboration with Ronnie Pavlov.
Existence of a measurable compensation function between subshifts and its application
We show the existence of a bounded Borel measurable compensation function for a factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set
of an expanding nonconformal map on the torus given by an
integer-valued diagonal matrix. We give a formula for the Hausdorff dimension for a compact invariant set represtended by a subshift and characterize the invariant ergodic measures of full dimension. We also study uniqueness of the measure of full dimension.
Elliptic periodic orbit for generic geodesic flow on the 2-sphere
Physical measures for infinite modal maps
We analyze certain parametrized families of one-dimensional maps with infinitely many critical points from the measure-theoretical point of view. We prove that Duch families have absolutely continuous invariant probability measures for a positive Lebesgue measure subset of parameters. Moreover we show that both the densities of these measures and their entropy vary continuously with the parameter. In addition we obtain exponential rate of mixing for these measures and also that they satisfy the Central Limit Theorem.
Dynamical Teichmuller spaces
We discuss basic properties of the space of local qc deformations of a rational map R. In particular, we show that if R is a polynomial whose Julia set is homeomorphic to a Cantor set, then the space local qc deformations has a laminated structure. This is a Joint work with Petr Makienko.
Linearly repetitive Delone sets: tower systems and patch frequencies
C2 – Robust transitivity in surfaces with boundary
Robustly transitive maps can not be ignored in any global picture of dynamical systems. Besides being a property that cannot be destroyed by small perturbations, the maps that have it often exhibit a chaotic dynamical behavior. This property is widely studied in the C1-topology, and it is related to hyperbolicity and transversalita properties. Few things are known in the C2-topology. In this talk we shall exhibit a class of smooth diffeomorphisms on a compact surface with boundary which are robustly transitive in the C2-topology but not in the C1-topology.
Robust expansiveness and entropy-expansiveness implications at the infinitesimal level
Let  be a Cr-diffeomorphism,  defned on a compact boundaryless -dimensional manifold , , and let  be the homoclinic class associated to the hyperbolic periodic point p. We analyze the consequences in the behavior of the tangent map of a robust property. In particular we focuses in the cases of C1- robust h-expansiveness and robust expansiveness proving that in that cases there exist a dominated splitting in the case of robust h-expansiveness and for the case of robust expansiveness hyperbolicity holds generically.
If dominance implies h-expansiveness, the converse implication, is considered too.